Lesson 3: Graphing with Intercepts

New Concept

Discover a shortcut for graphing lines! We'll learn to identify intercepts—the points where a line crosses the $x$- and $y$-axes. You'll master finding these from an equation and using them to quickly and accurately draw the line.

What’s next

You've got the foundation. Now, you’ll tackle a series of interactive examples and practice problems to master graphing with intercepts.

Property

Each of the points at which a line crosses the $x$-axis and the $y$-axis is called an intercept of the line.

The $x$-intercept is the point, $(a, 0)$, where the graph crosses the $x$-axis. The $x$-intercept occurs when $y$ is zero.

The $y$-intercept is the point, $(0, b)$, where the graph crosses the $y$-axis. The $y$-intercept occurs when $x$ is zero.

Property

Use the equation to find:
• the $x$-intercept of the line, let $y = 0$ and solve for $x$.
• the $y$-intercept of the line, let $x = 0$ and solve for $y$.

Examples

• For the equation $x + 4y = 8$: To find the x-intercept, let $y=0$, which gives $x=8$. The x-intercept is $(8,0)$. For the y-intercept, let $x=0$, which gives $4y=8$, so $y=2$. The y-intercept is $(0,2)$.

• For the equation $6x - 2y = 12$: Let $y=0$, so $6x=12$ and $x=2$. The x-intercept is $(2,0)$. Let $x=0$, so $-2y=12$ and $y=-6$. The y-intercept is $(0,-6)$.

Property

Step 1. Find the $x$- and $y$-intercepts of the line.
• Let $y = 0$ and solve for $x$.
• Let $x = 0$ and solve for $y$.

Step 2. Find a third solution to the equation.

Step 3. Plot the three points and then check that they line up.

Property

Step 1. If the equation has only one variable. It is a vertical or horizontal line.
• $x = a$ is a vertical line passing through the $x$-axis at $a$.
• $y = b$ is a horizontal line passing through the $y$-axis at $b$.

Step 2. If $y$ is isolated on one side of the equation. Graph by plotting points.
• Choose any three values for $x$ and then solve for the corresponding $y$-values.

Step 3. If the equation is of the form $Ax + By = C$, find the intercepts.
• Find the $x$- and $y$-intercepts and then a third point.

The Strategy

Not all linear equations are best graphed the same way. By looking at the form of the equation, you can choose the quickest and easiest method. Here’s a simple strategy:

1. One Variable? It's a Vertical or Horizontal Line.

• If the equation is $x = a$, it's a vertical line that passes through the $x$-axis at the point $(a, 0)$.
• If the equation is $y = b$, it's a horizontal line that passes through the $y$-axis at the point $(0, b).

2. Is $y$ Alone? Use Slope-Intercept Form.

• If the equation is in slope-intercept form ($y = mx + b$), the easiest method is to plot points.
• Pick three simple values for $x$ (like 0, 1, and a multiple of the slope's denominator) and find the corresponding $y$-values.